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Theorem nfsum1 10331
Description: Bound-variable hypothesis builder for sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)
Hypothesis
Ref Expression
nfsum1.1  |-  F/_ k A
Assertion
Ref Expression
nfsum1  |-  F/_ k sum_ k  e.  A  B

Proof of Theorem nfsum1
Dummy variables  f  j  m  n  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-isum 10329 . 2  |-  sum_ k  e.  A  B  =  ( iota x ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m
)  /\  A. j  e.  ( ZZ>= `  m )DECID  j  e.  A  /\  seq m
(  +  ,  ( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) ,  CC )  ~~>  x )  \/  E. m  e.  NN  E. f
( f : ( 1 ... m ) -1-1-onto-> A  /\  x  =  (  seq 1 (  +  ,  ( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n
)  /  k ]_ B ,  0 ) ) ,  CC ) `
 m ) ) ) )
2 nfcv 2220 . . . . 5  |-  F/_ k ZZ
3 nfsum1.1 . . . . . . 7  |-  F/_ k A
4 nfcv 2220 . . . . . . 7  |-  F/_ k
( ZZ>= `  m )
53, 4nfss 2993 . . . . . 6  |-  F/ k  A  C_  ( ZZ>= `  m )
63nfcri 2214 . . . . . . . 8  |-  F/ k  j  e.  A
76nfdc 1590 . . . . . . 7  |-  F/ kDECID  j  e.  A
84, 7nfralxy 2403 . . . . . 6  |-  F/ k A. j  e.  (
ZZ>= `  m )DECID  j  e.  A
9 nfcv 2220 . . . . . . . 8  |-  F/_ k
m
10 nfcv 2220 . . . . . . . 8  |-  F/_ k  +
113nfcri 2214 . . . . . . . . . 10  |-  F/ k  n  e.  A
12 nfcsb1v 2939 . . . . . . . . . 10  |-  F/_ k [_ n  /  k ]_ B
13 nfcv 2220 . . . . . . . . . 10  |-  F/_ k
0
1411, 12, 13nfif 3385 . . . . . . . . 9  |-  F/_ k if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 )
152, 14nfmpt 3878 . . . . . . . 8  |-  F/_ k
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) )
16 nfcv 2220 . . . . . . . 8  |-  F/_ k CC
179, 10, 15, 16nfiseq 9528 . . . . . . 7  |-  F/_ k  seq m (  +  , 
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) ,  CC )
18 nfcv 2220 . . . . . . 7  |-  F/_ k  ~~>
19 nfcv 2220 . . . . . . 7  |-  F/_ k
x
2017, 18, 19nfbr 3837 . . . . . 6  |-  F/ k  seq m (  +  ,  ( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) ,  CC )  ~~>  x
215, 8, 20nf3an 1499 . . . . 5  |-  F/ k ( A  C_  ( ZZ>=
`  m )  /\  A. j  e.  ( ZZ>= `  m )DECID  j  e.  A  /\  seq m (  +  , 
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) ,  CC )  ~~>  x )
222, 21nfrexya 2406 . . . 4  |-  F/ k E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  A. j  e.  ( ZZ>= `  m )DECID  j  e.  A  /\  seq m (  +  , 
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) ,  CC )  ~~>  x )
23 nfcv 2220 . . . . 5  |-  F/_ k NN
24 nfcv 2220 . . . . . . . 8  |-  F/_ k
f
25 nfcv 2220 . . . . . . . 8  |-  F/_ k
( 1 ... m
)
2624, 25, 3nff1o 5155 . . . . . . 7  |-  F/ k  f : ( 1 ... m ) -1-1-onto-> A
27 nfcv 2220 . . . . . . . . . 10  |-  F/_ k
1
28 nfv 1462 . . . . . . . . . . . 12  |-  F/ k  n  <_  m
29 nfcsb1v 2939 . . . . . . . . . . . 12  |-  F/_ k [_ ( f `  n
)  /  k ]_ B
3028, 29, 13nfif 3385 . . . . . . . . . . 11  |-  F/_ k if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  0 )
3123, 30nfmpt 3878 . . . . . . . . . 10  |-  F/_ k
( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  0 ) )
3227, 10, 31, 16nfiseq 9528 . . . . . . . . 9  |-  F/_ k  seq 1 (  +  , 
( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  0 ) ) ,  CC )
3332, 9nffv 5216 . . . . . . . 8  |-  F/_ k
(  seq 1 (  +  ,  ( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n
)  /  k ]_ B ,  0 ) ) ,  CC ) `
 m )
3433nfeq2 2231 . . . . . . 7  |-  F/ k  x  =  (  seq 1 (  +  , 
( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  0 ) ) ,  CC ) `
 m )
3526, 34nfan 1498 . . . . . 6  |-  F/ k ( f : ( 1 ... m ) -1-1-onto-> A  /\  x  =  (  seq 1 (  +  ,  ( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n
)  /  k ]_ B ,  0 ) ) ,  CC ) `
 m ) )
3635nfex 1569 . . . . 5  |-  F/ k E. f ( f : ( 1 ... m ) -1-1-onto-> A  /\  x  =  (  seq 1 (  +  ,  ( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n
)  /  k ]_ B ,  0 ) ) ,  CC ) `
 m ) )
3723, 36nfrexya 2406 . . . 4  |-  F/ k E. m  e.  NN  E. f ( f : ( 1 ... m
)
-1-1-onto-> A  /\  x  =  (  seq 1 (  +  ,  ( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n
)  /  k ]_ B ,  0 ) ) ,  CC ) `
 m ) )
3822, 37nfor 1507 . . 3  |-  F/ k ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  A. j  e.  (
ZZ>= `  m )DECID  j  e.  A  /\  seq m
(  +  ,  ( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) ,  CC )  ~~>  x )  \/  E. m  e.  NN  E. f
( f : ( 1 ... m ) -1-1-onto-> A  /\  x  =  (  seq 1 (  +  ,  ( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n
)  /  k ]_ B ,  0 ) ) ,  CC ) `
 m ) ) )
3938nfiotaxy 4901 . 2  |-  F/_ k
( iota x ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m
)  /\  A. j  e.  ( ZZ>= `  m )DECID  j  e.  A  /\  seq m
(  +  ,  ( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) ,  CC )  ~~>  x )  \/  E. m  e.  NN  E. f
( f : ( 1 ... m ) -1-1-onto-> A  /\  x  =  (  seq 1 (  +  ,  ( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n
)  /  k ]_ B ,  0 ) ) ,  CC ) `
 m ) ) ) )
401, 39nfcxfr 2217 1  |-  F/_ k sum_ k  e.  A  B
Colors of variables: wff set class
Syntax hints:    /\ wa 102    \/ wo 662  DECID wdc 776    /\ w3a 920    = wceq 1285   E.wex 1422    e. wcel 1434   F/_wnfc 2207   A.wral 2349   E.wrex 2350   [_csb 2909    C_ wss 2974   ifcif 3359   class class class wbr 3793    |-> cmpt 3847   iotacio 4895   -1-1-onto->wf1o 4931   ` cfv 4932  (class class class)co 5543   CCcc 7041   0cc0 7043   1c1 7044    + caddc 7046    <_ cle 7216   NNcn 8106   ZZcz 8432   ZZ>=cuz 8700   ...cfz 9105    seqcseq 9521    ~~> cli 10255   sum_csu 10328
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-un 2978  df-in 2980  df-ss 2987  df-if 3360  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-mpt 3849  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-recs 5954  df-frec 6040  df-iseq 9522  df-isum 10329
This theorem is referenced by: (None)
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